The study of automorphic forms is at the center of modern number theory. As mentioned previously they are a natural generalisation of the notion of modular form.
In order to motivate the ideas we will observe the rite of passage involved in promoting elliptic modular forms to automorphic forms. It is possible to carry out similar procedures for other common types of modular form but the details are not important to us here.
The approach here will be mainly taken from Chapter 7 of [7] although we will only see a brief overview of the theory.
To start the process we take an elliptic modular form f ∈ Sk(Γ0(N )). Thus
f is a holomorphic function H → C with specific transformation properties. The action of GL+2(R) = {A ∈ GL2(R) | det(A) > 0} on H is transitive
as seen in Chapter 1. The element i ∈ H is special since the stabilizer of this element under the action is easily described, it is K∞ = Z∞K0, where
Z∞ := {diag(z, z) | z ∈ R×} is the center of GL+2(R) and K0 = SO2(R) is a
maximal connected compact subgroup of GL+2(R).
We may now associate to f a smooth function Φf on GL+2(R) by “undoing”
the action of the automorphy factor at i:
Φf : GL+2(R) −→ C
Φf(g∞) = j(g∞, i)−kf (g∞i).
Here the automorphy factor on GL+2(R) is extended from SL2(R) by defining it
as j(γ, z) = det(γ)−12(cz + d) for γ =
a b
c d
∈ GL+2(R). This clearly agrees
with the usual automorphy factor if we restrict to γ ∈ SL2(R).
Note that from Φf we may recover f via f (z) = j(g∞, i)kΦ(g∞) where
g∞ ∈ GL+2(R) is such that z = g∞i (this exists by the transitivity of the
action).
The following are simple consequences of the transformation properties of f . Theorem 3.1.1. The function Φf has the following properties:
• Φf(γg∞) = Φf(g∞) for each γ ∈ Γ0(N ).
• Φf(gzkθ) = Φf(g∞)eikθ for each zkθ∈ K∞.
The first of these properties tells us that φf is well defined as a function on
3.1. Classical automorphic forms
the level of the modular form f . The second property tells us how the weight of f is encoded in the action of the maximal connected compact subgroup K0.
It remains to see how we may capture the holomorphicity of f algebraically. First note that the Lie algebra gl2= M2(R) of GL+2(R) acts on smooth functions
Φ : GL+2(R) → C. For X ∈ M2(R) this action is as follows:
X · Φ(g∞) =
d
dt(Φ(g∞exp(tX)))t=0.
By extending this action linearly we can allow the action of the complexifi- cation g = gl2⊗ C. Two special elements of this Lie algebra are:
X±= 1 2 1 ±i ±i −1 ∈ g.
The action of these elements raise and lower the weight, i.e. for kθ ∈ K0 we
have:
X±· Φ(g∞kθ) = Φ(g∞)eiθ(k±2).
The following intriguing result is the sought after analogue of holomorphicity, linking the action of X− with the Cauchy-Riemann equations.
Lemma 3.1.2. The modular form f is holomorphic if and only if X−· Φf = 0.
So now we have embedded Sk(Γ0(N )) into a space of functions GL+2(R) → C
with the properties given in the above theorem and lemma.
Indeed one can study these spaces and produce nice results about modular forms but we are not quite finished yet. At the moment we are only focusing on a single completion of Q but experience tells us that we should be considering all completions, i.e. we would like to lift to a function on the adelic group GL2(A).
In order to do so we must first think about what the analogue of Γ0(N )
would be. What we require is a subgroup of GL2(Qp) for each p that locally
behaves like Γ0(N ). For any prime p we may make the obvious choice:
Kp(N ) = a b c d ∈ GL2(Zp) c ∈ pordp(N )Zp ⊂ GL2(Qp).
Note that if p - N then we have Kp(N ) = GL2(Zp) and so it makes sense
to say that Kf(N ) = QpKp(N ) is a subgroup of GL2(Af). In fact it is an
open compact subgroup. We also consider the open compact subgroup K(N ) = K∞Kf(N ) of GL2(A).
Recall the following decomposition of A×:
where ˆZ× = Q
pZ ×
p. This is the adelic analogue of the Chinese remainder
theorem and allows us to lift Dirichlet characters to continuous homomorphisms Q×\A×→ C.
One has a similar theorem for GL2(A) and this result will allow us to achieve
our goal of lifting Φf to a function on GL2(A).
Theorem 3.1.3. (Strong approximation) Suppose Kf ⊂ GL2(Af) is any open
compact subgroup with det(Kf) = ˆZ×. Then we have the decomposition: GL2(A) = GL2(Q)GL+2(R)Kf.
Thus:
GL2(Q)\GL2(A)/Kf ∼= Γ\GL+2(R),
where Γ = GL+2(Q) ∩ Kf.
In the case where Kf = Kf(N ) we recover Γ0(N ) = GL+2(Q) ∩ Kf(N ) (this
is probably easiest to see for N = 1).
Note that the strong approximation theorem gives us an adelic version of the modular curve Γ\H (the fundamental domain for the action of Γ on the upper half plane H). Indeed quotienting further:
GL2(Q)\GL2(A)/K∞Kf∼= Γ\ GL2+(R)/K∞∼= Γ\ (SL2(R)/SO2(R)) ∼= Γ\H.
If we consider this double coset instead over GL2(Af) then we get a simple
space.
Lemma 3.1.4. |GL2(Q)\GL2(Af)/Kf| is finite (in fact if Kf is as in the strong
approximation theorem then the size is 1).
This result still holds for any connected reductive group G in place of GL2
and any open compact subgroup Kf ⊆ G(Af) (although strong approximation
can fail due to the double quotient being non-trivial). However fixing represen- tatives gi∈ G(Af) for G(Q)\G(Af)/Kf the analogue of Theorem 3.1.3 is:
G(Q)\G(A)/Kf∼=
a
i
Γi\G(R),
where Γi= G(Q) ∩ gi−1Kfgi.
Letting K∞ = Z(R)K0 where Z is the center of G and K0 is a max-
imal connected compact subgroup of G(R) we then get a decomposition of G(Q)\G(A)/K∞Kf into a disjoint product of locally symmetric spaces. The
role of the upper half plane H ∼= SL2(R)/SO2(R) is now given by a connected
3.1. Classical automorphic forms
We tend to find arithmetic data in the double coset G(Q)\G(Af)/Kf. Later
we will see connections to the genus theory of lattices. For now we give its size an appropriate name.
Definition 3.1.5. When the choice of reductive G and open compact Kf is
understood the number h = |G(Q)\G(Af)/Kf| is referred to as the class num-
ber.
Example 3.1.6. As mentioned earlier, for G = GL2 or G = SL2 and any
suitable choice of Kf (for example ones corresponding to congruence subgroups)
we have class number 1.
The reason for the use of the term class number is not coincidental. Let F be any number field with ring of integers OF and choose an ideal m ⊆ OF.
Consider the reductive group G = Gm= GL1 and open compact Kf =QpKp
with Kp= 1 + mOF,p for each prime ideal p ⊂ OF. Then:
G(F )\G(AF,f)/Kf = IF,f/F×Kf,
where IF,f = A×F,f are the finite ideles of F .
This matches the definition of idele ray class groups for moduli with non- archimedean part m (see Chapter 3 of [15]). In particular for m = OF we recover
the ideal class group of F and so the notion of class number here is really the size of G(F )\G(Af)/Kf.
To summarize, so far we have taken f ∈ Sk(Γ0(N )) and produced a function
on Γ\GL+2(R) with nice properties. Using strong approximation we may now produce a function Φf : GL2(A) → C via:
Φf(g) = Φf(γg∞k) = Φf(g∞),
where g ∈ GL2(A), γ ∈ GL2(Q), g∞∈ GL+2(R), k ∈ Kf(N ).
We now have the following result:
Theorem 3.1.7. The map f 7→ Φf is an isomorphism from Sk(Γ0(N )) to the
space of functions GL2(A) → C satisfying:
• Φf(γgk) = Φf(g) for all γ ∈ GL2(Q) and k ∈ Kf(N ).
• The function g∞ 7→ Φf(g∞gf) is smooth for any gf ∈ GL2(Af) and sat-
isfies the properties of Theorem 3.1.1 and Lemma 3.1.2. • The function Φf is cuspidal, i.e:
Z Q\A Φf 1 x 0 1 g dx = 0 for all g ∈ GL2(A).
We will not explain how the third property translates cuspidality of f into this integral condition. A justification is found on p.137 − 138 of [7].
Definition 3.1.8. The function Φf is the automorphic form associated to f .
We note that there are many generalisations of the automorphic forms con- structed above.
• One can define automorphic forms for other open compact subgroups by using the exact same process as above.
• Given a character ω : Q×\A× → C× we may modify the definition of
automorphic form to demand that the center acts by this character, i.e. φf(gz) = ω(z)f (g) for all z ∈ Z∞. Such automorphic forms are said to
have central character ω.
In this fashion modular forms with Dirichlet character χ lift to give au- tomorphic forms with central character ωχ (where ωχ is the lift of χ by
using strong approximation for A×).
• One may define automorphic forms for other number fields, taking care of the possibility that more archimedean places may exist.
• We may define automorphic forms for other reductive groups in place of GL2.
As a final remark we also note that not all automorphic forms are attached to modular forms. I refer the reader to p.138 − 139 of [7] for a general definition.